Efficient simulation of rare jitter probabilities in ATM switches

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E cient Simulation of Rare Jitter Probabilities in

ATM Switches

Ahmet A. Akyamac J. Keith Townsend

Center for Advanced Computing and Communication North Carolina State University

Raleigh, N.C.

Abstract

The ATM Forum has dened two dierent types of cell delay variation (CDV) measures: The 1-Point CDV, which is a measure of jitter, is typically specied for constant bit rate (CBR) sources (eg. real-time video transfer) and the 2-Point CDV can be specied for both CBR and variable bit rate (VBR) sources (eg. compressed video). Analytical results are not available and the events associated with large jitter in ATM networks are typically rare (< 10;6), hence Monte Carlo simulation is not feasible. In this paper, we model trac using

the ATM Forum standardized trac descriptors and consider dispersion as a measure of jitter. We extend our previous work on remote delay quantiles for heterogeneous systems to generate ecient simulation techniques using Importance Sampling (IS) to estimate the 2-Point CDV for mixed CBR and VBR sources. Subsequently, we present a novel IS simulation methodology to estimate the 1-Point CDV for CBR sources in the presence of background VBR trac. For both cases, we observe from experimental results that the improvement in simulation eciency is inversely proportional to the probability being estimated1_.

1This work was supported by the Center for Advanced Computing and Communication, North Carolina

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 1

1 Introduction

Cell Delay Variation (CDV), or jitter, is an important Quality of Service (QoS) measure that will be provided in Asynchronous Transfer Mode (ATM) Networks. The ATM Forum 1] has dened two dierent types of CDV measures: The 1-Point CDV is typically specied for constant bit rate (CBR) sources (eg. real-time video transfer) and the 2-Point CDV can be specied for both CBR and variable bit rate (VBR) sources (eg. compressed video). The events associated with large jitter in ATM networks are typically rare, on the order of 10;9

and below.

Rather than using statistical models to characterize the input trac, in this paper we use the connection trac descriptors standardized by the ATM Forum. These descriptors are the peak cell rate ^ in Mbps, the mean cell rate in Mbps and the maximum burst duration ^B cells at the peak rate. This approach has been called the operational approach in 2].

Due to the low probabilities involved, large buer sizes and the operational denition of the input trac, accurate analytical solutions to CDV are unavailable. Thus, simulation is required to estimate the CDV probabilities.

Various forms of cell delay variation have been considered before in the literature. In 3], the waiting time distributions for VBR sources are analyzed at the output of an ATM switch, where the service time is assumed to be identical to the transmission time of a packet. The departure process of VBR cells is also considered in the context ofMMPP=D=1 and M=D=1 queues in 4]. But, neither of these works considers CBR trac with VBR background trac. In 5], jitter is dened in terms of tail probabilities of the queueing delay and also in terms of the squared coecient of variation of the intercell delay at the output. The CBR trac is assumed to be multiplexed with Poisson background trac. The trac model in 6] is the same as the one on 5] and it is assumed that the sequence of random delays for the CBR cells constitutes a Markov process. The Poisson background trac assumption is not realistic in the case of bursty VBR sources, which are present in ATM networks. In 7], the VBR trac is modeled as a superposition of 2-state Markov processes and the CBR trac is modeled as a special case of a Markovian D;BMAP process. Approximate solution techniques are

given for the tail probabilities of the cell delays. The ATM Forum trac descriptors are used in 8] to generate tail probabilities, but the service time is assumed to be identical to the transmission time of a cell. The tail distribution described above corresponds to the 2-Point CDV denition. In 9], the 1-Point CDV denition is considered using comparisons between the interdeparture times and the CBR period. The background VBR trac is in the form of independent and identically distributed batches of customers.

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1 MP2

MP

CBR

network

T <T >T

background VBR

Figure 1: Cell Delay Variation through a network.

probabilities, we use Importance Sampling (IS) as a means of generating ecient simula-tions. For the 2-Point CDV results, we extend the techniques we used previously in 10] for the case of aggregate delay threshold probabilities with heterogeneous trac. For the 1-Point CDV results, we present, for the rst time, a new IS procedure to generate ecient simulations.

Cell delay variation for CBR trac in ATM networks can manifest itself as either cell clumping or cell dispersion 11]. In this paper, we focus on cell dispersion as a quality of service measure. Cell clumping is bounded by the period of the CBR source, whereas cell dispersion can occur with signicantly higher probability than cell clumping. At the expense of large buers and induced delay, cell clumping can be eectively reduced by the use of jitter removal buers 12]. However, it may not be possible to cancel the eects of very large delays in the network, in which case cells which are very late would be dropped. For this reason, it is important to characterize probabilities for cell dispersion.

This paper is organized as follows: Section 2 gives a description of the two ATM Forum CDV denitions. The system description and simulation setup are presented in Section 3. The ecient simulation methodology for the 2-Point CDV is described in Section 4. A detailed development of the IS simulation method for the 1-Point CDV follows in Section 5. Experimental results are presented in Section 6, followed by conclusions in Section 8.

2 The ATM Forum CDV Denitions

For CBR trac that is not distorted by cell delay variation, the interarrival times of all CBR cells are constant and are given byTCBR. However, when these CBR cells pass through

a network (which contains switches, buers and other background trac) as in Fig. 1, the interdeparture times are no longer constant and may vary signicantly from TCBR.

In-terdeparture times less than TCBR refer to clumping and those greater than TCBR refer to

dispersion. Cell delay variation, or jitter, is a measure of how signicant a change is induced on the instantaneous interdeparture times of the CBR cells. In essence, this change will result in interdeparture times dierent than TCBR. This, in turn, may have severe eects

depending on the application using the CBR cells. For example, for a 1.5 Mbps MPEG NTSC video source, a delay variation of more than 11 ms: is unacceptable 13].

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 3 CDV measures jitter at a single measurement point (MP). In Fig. 1, MP2 would be such

a point. The two-point CDV measures jitter between two MP's, such as MP1 and MP2 in

Fig. 1.

Following the notation in 1], the one-point CDV for cellk, yk, at a measurement point,

say MP2, is dened as the dierence between the cell's reference arrival time, ck, and the

cell's actual arrival time at MP2, ak, and is thus given byyk =ck;ak. In this formulation,

the reference arrival times are computed as follows: c0 = a0 = 0

ck+1 =

(

ck+TCBR if ck ak

ak +TCBR otherwise (1)

Positive values of CDV correspond to cell clumping (interdeparture times less then TCBR)

and negative values of CDV correspond to cell dispersion (interdeparture times greater than TCBR). The one-point CDV is typically used for CBR sources.

We mentioned in Section 1 that we focus on dispersion as a CDV measure. Thus, the important events are characterized by cells for whichyk <;, where is the target dispersion

value, which we specify as a percentage of the CBR period TCBR.

The two-point CDV for cell k, vk, between two measurement points MP1 and MP2 is

dened as the dierence between the absolute cell transfer delay of cell k, xk, between the

two reference pointsMP1 andMP2 and a reference cell transfer delay,d1 2betweenMP1 and

MP2 and is thus given by vk =xk;d1 21]. The reference cell transfer delay is the absolute

delay between MP1 and MP2 experienced by a reference cell. Note that the absolute cell

transfer delay is nothing but the Cell Transfer Delay dened in 1] and hence can be used for both CBR and VBR sources. The important events for the two-point CDV are characterized by cells for which vk > , where is the target delay value.

3 Simulation Setup

3.1 System Description

The model we use for the ATM switch is given in Fig. 2. The switch hasNP input ports and

NP output ports. We assume that the number of connections isNC =NP. Each connection

is routed uniformly and instantaneously to one of the output ports through the nonblocking bus. Due to the uniform nature of the trac, we analyze a single tagged output buer to represent the performance of all output buers. Also, for a worst case analysis, we assume that allNC connections are routed to the tagged output buer. Even though we specify the

output buers to have a length of K, we will assume that they have innite capacity and we will use K as a parameter for the two-point CDV calculations. The buers are served at a constant rate of ^ using a FIFO queueing discipline. We place no restriction on the value of ^ or the relative values of ^ and the trac rates.

We assume there are two classes of trac: CBR trac has the triplet (^CBR CBR ^BCBR)

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µ

^

µ

^

µ

^

µ

VBR

^

µ

^

µ

^

MP

1 MP2

tagged bus

CBR

Figure 2: The ATM Switch Model.

CBR and ^BCBR = 1. We assume that ^VBR = k^CBR, where k 2 Z+ and k 2. For the

VBR class, we make the worst case assumption that the trac arrives according to thegreedy

pattern. In the ON period, the VBR source generates trac at the peak rate of ^VBR for a

burst duration of ^BVBR, and the OFF period is required to average out to the mean rate of

VBR.

3.2 Simulation Procedure

We use a slotted-time simulation model resulting from the normalization with respect to the peak rate of the VBR source, ^VBR. In the normalized model, one slot corresponds to

the transmission of a single 53-byte ATM cell at the rate of ^VBR. After normalization,

the equivalent arrival rate of the VBR source is 1 cell/slot and that of the CBR source is ^CBR=^VBR = 1=k cells/slot, where k was dened above. Due to the greedy trac

assump-tion, the VBR source is periodic. By deniassump-tion, the CBR source is also periodic. After normalization, the periods are given by:

TVBR = ^VBR_VBR^BVBR (2)

TCBR = ^VBR ^BCBR CBR = k

^CBR

CBR =k (3)

Since both the VBR and CBR trac are periodic, the entire input trac has a super-period of TSP= lcm(TVBRTCBR). We assumeTVBR to be an integer multiple ofk so that TVBR=TSP.

After normalization, the equivalent service rate is given by = ^=^VBR cells/slot. An

example of the resulting slotted-time pattern is plotted in Fig. 3 for ^VBR = 6 cells/slot,

VBR = 1 cell/slot, ^BVBR = 5 cells and CBR = ^CBR = 1 cell/slot and ^BCBR = 1 cell where

both connections start at slot 0.

The shared bus switch architecture results inservice slotswithin an arrival slot, as shown in Fig. 4. At an arrival slot, there can be up to NC cell arrivals and thus the number of

service slots per arrival slot is NC. We make a worst case assumption that if at an arrival

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 5 29 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 000 000 111 111 00 00 11 11 00 00 11 11 00 00 11 11 0

Figure 3: Slotted-time example pattern for CBR/VBR trac.

• • • 1 2 N_C Cell Arrivals • • • Arrival Slot Boundary Service Slot Arrival Slot Boundary Arrival Slot

1 2 N_C

Figure 4: Arrival slot and service slot relationship.

slots. In each service slot, one cell can be loaded into the output buer and =NC cells can

be serviced.

The cells exit the buer in slotted-time, where one departure slot is equivalent to one arrival slot. If cell x arrives at arrival slot ax at line sx (which is, by denition, the service

slot) and encounters a queue length of qx upon arrival, then the departure slot for cell x is

given by:

dx =ax+

2 6 6 6

qx+1

=NC +sx;1

NC

3 7 7 7

;1 (4)

The randomness in the simulation is found in the connection starting-slots. Each of the NVBR connections can be characterized by a discrete-time random variable, uniform on

0TSP ; 1] representing the starting-slot of the VBR connection. Similarly, each of the

NCBR CBR connections can be characterized by a discrete-time random variable, uniform

on 0TCBR ;1]. Here, NC = NCBR +NVBR. We represent the starting-slots as an NC

-dimensional vectorv, which can be partitioned into two vectors vCBR andvVBR representing

the starting-slots of the CBR and VBR connections, respectively, as follows:

v = (vVBRjvCBR) (5)

4 E cient Simulation of the Two-Point CDV

We mentioned in Section 2 that the two-point CDV involves the absolute cell transfer delay vk for cellk and that we focus on the events for which vk > , where is a reference absolute

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 6 probability to be the probability that the end to end delay through the switch exceeds a given threshold , Pr(D > ), where represented a percentage of the specied buer length K.

Here, we assume that MP1 is the entry point into the switch andMP2 is the exit point

from the switch. Thus, in this context, the two-point CDV is identical to the delay threshold probability considered in 10]. Since we assumed that the routing within the switch is instantaneous, the delay between pointsMP1 andMP2is a result of the queueing. We use the

techniques developed in 10] to generate the two-point CDV for the individual classes. Note that we consider the two-point CDV for both the CBR and VBR classes. For consistency, we use the notation in 10] whereby we represent the two-point CDV (or delay threshold probability) using PTH.

Following the development in 10], letV be the set of all connection starting-slot vectors and nTHj be the number of connection starting-slot vectors that map to exactlyj cells that

exceed the threshold in steady state. Here, jVj = (TVBR)NVBR (TCBR)NCBR, where jj

denotes the cardinality of the set. The probability that j cells exceed the threshold in steady state is given bypTHj =nTHj=jVjforj = 0Dmax, whereDmax is the maximum number

of cells that can exceed the threshold. Note that P_D_max

j=0 pTHj = 1. The average number of

cells nTH that exceed the threshold in steady state is then given by nTH = PDj=0maxjpTHj.

Hence, the delay threshold probability for the innite buer is given by: PTH =DXmax

j=0

_j

Ncells

pTHj (6)

where Ncells is the total number of cells that arrive in a superperiod,

Ncells=NVBR^BVBR+NCBR

TSP

^VBR=^CBR (7)

The above formulation expresses PTH as a weighted sum of multinomial probabilities and

thus creates a \multiple bin" probability structure composed of bins numbered 0 to Dmax,

where bin j corresponds to j cells that exceed the threshold. The value of Dmax can be

found using a single simulation run 10].

Calculating the exact delay threshold probability has a complexity of O((TVBR)NVBR

(TCBR)NCBR) and is thus intractable for realistic systems. Monte Carlo simulation is not

feasible due to the very low probabilities involved (typically in the range of 10;9 to 10;12. In

14, 10], we developed a three-part Importance Sampling simulation procedure to eciently estimate PTH. The rst two parts identied a set V which contained all of the vectors

that caused a cell to exceed the delay threshold and generated the related IS weights. The third part took advantage of the multiple bin structure to further enhance the simulation eciency.

Using the simulation procedure in 10], the aggregate delay threshold probability is esti-mated as follows:

^P

TH =DXmax j=1

j

Ncells

^ p

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 7 where ^p

THj are the estimates for the individual bins and are given by:

^ p

THj = 1n_j

nj

X

s=1I

j(s)w

j(s) (9)

where I

j(s) is the indicator function for j cells exceeding the threshold at run s, w

j(s) is a

vector dependent IS weight function and nj is the number of simulation runs for bin j. We

estimate the variance of the IS estimator as follows: ^2_(^_p

THj) = n_j(n_j1;1)

nj

X

s=1(I

j(s)w

j(s);p^

THj)2 (10)

and following 14, 10], we generate condence intervals using the result that the condence interval of a weighted sum of individual multinomial probabilities follows a 2 _{distribution.}

The improvement in simulation eciency is given by ^Rnet =

P_D_max

j=1 j2p^

THj(1;p^

THj)=nj

P_D_max

j=1 j2^2(^p

THj) (11)

Note that the two-point CDV, or delay threshold probabilities for the individual classes, PCBR and PVBR are given by:

PCBR = 1_C_CBR DXmax

j=1 pCBRj (12)

PVBR = _C_VBR1 DXmax

j=1 pVBRj (13)

whereCCBR =NCBRTSP^CBR=^VBR is the total number of CBR cells andCVBR=NVBR^BVBR

is the total number of VBR cells. Here, pCBRj is the expected number of CBR cells that

exceed the threshold conditioned on a total of j cells that exceed the threshold and pVBRj

is the expected number of VBR cells that exceed the threshold conditioned on a total of j cells that exceed the threshold. It is very dicult to identify sets of vectors that will result in exactly k CBR and l VBR cells that exceed the threshold. For this reason, we use the simulation biasing procedure in 10] to identify the set V that contains all vectors which

cause at least one cell to exceed the threshold. Hence, all vectors which cause CBR and VBR cells to exceed the threshold will also be included in this set.

For the CBR class, the estimate of the two-point CDV, ^P

CBR, is given by:

^P

CBR = 1_C_CBR DXmax

j=1 p^

CBRj (14)

where ^p

CBRj is generated as follows:

^ p

CBRj = 1n

j nj

X

s=1Mjs(CBR)w

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 8 Here, w

j(s) is the IS weight dened earlier and Mjs(CBR) is the number of CBR cells that

exceed the threshold at run s, given that at total of j cells exceed the threshold. Similarly, the estimate of the two-pint CDV for the VBR class, ^P

VBR, is given by:

^P

VBR= 1_C_CBR DXmax j=1 p^

VBRj (16)

where ^p

VBRj is generated as follows:

^ p

VBRj = 1n

j nj

X

s=1Mjs(VBR)w

j(s) (17)

For this case,Mjs(VBR) is the number of VBR cells that exceed the threshold at run s, given

that at total of j cells exceed the threshold.

Hence, to estimate the individual delay threshold probabilities (two-point CDV), we run IS simulations as in 10] and collect the aggregate statistics as well as individual CBR and VBR statistics. Since the biasing is performed according to the aggregate probability, con-dence intervals and improvement factors are only calculated for the aggregate estimate. Note that if the CBR and VBR classes were targeted individually, the individual improvements would be only slightly higher. But, as mentioned before, it is very dicult to target the IS biasing for individual classes. Furthermore, since the set V contains all the vectors that

case at least one cell to exceed the threshold, the biasing procedure is valid for the individual classes and the individual estimates are unbiased. Experimental results for the two-point CDV are reported in Section 6.

5 E cient Simulation of the One-Point CDV

5.1 Motivation

For the one-point CDV, we assume that the measurement point is at the output of the ATM switch, labeled MP2 in Fig. 2. We consider the one-point CDV for the CBR class only. We

assume that there is a single CBR source at the rst input line. All other input lines have VBR sources, constituting the background trac.

For the CBR source, dispersion between two cellsi and i + 1 is illustrated in Fig. 5. At the input of the switch, the interarrival time of cells i and i + 1 is TCBR slots, whereas the

interdeparture time is Wi+1 > TCBR.

For the celli+1 shown in Fig. 5, the ATM Forum one-point CDV denition presented in Section 2 will result inyi+1=TCBR;Wi+1. In Section 2, we noted that important (dispersive)

events are characterized by yi+1 <;. This is thus equivalent to saying Wi+1 > TCBR+.

Hence, the tail probability of dispersion in the one-point CDV sense is given by the following:

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cbr Wi+1>Tcbr

T

i switch

i

i+1 i+1

Figure 5: Illustration of dispersion between two cellsi and i + 1.

where W is a random variable denoting the interdeparture times of cells. However, W is a parameter dened at the output of the switch. To translate W to a parameter internal to the switch, we note that the queueing delay experienced by celli upon arrival to the output buer, Di, is given by:

D0 = 0

Di = Pik=0Wk ;iTCBR , i1 (19)

Hence, the interdeparture time between cellsi and i + 1 is given by:

Wi+1=Di+1;Di+TCBR (20)

Thus, the probability of dispersion can also be expressed as:

Pr(Di+1;Di > ) (21)

For conciseness, the probability of dispersion will also be represented byPDP in the remainder

of this paper.

5.2 Multinomial Formulation

In a period TSP, the total number of pairs of CBR cells is given by:

Npairs= T_T SP

CBR (22)

For a given set of parameters for the CBR and VBR classes and a given dispersion value , let Jmax be the maximum number of CBR pairs that are dispersed by more than (ie. that

have an interdeparture time greater thanTCBR+). Since there is only one CBR source and

also due to the periodicity of the system, we can assume, without loss of generality, that the CBR source starts at slot 0. Thus, the starting slot vectorv is given by:

v = (vVBRj0) (23)

As before, V is the set of all possible connection starting-slot vectors, where for this case,

jVj = (NVBR)TSP. Let nDPj be the number of connection starting-slot vectors that map to

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 10 j pairs are dispersed by more than is given by pDPj =nDPj=jVj for j = 0:::Jmax. Note

that P_J_max

j=0 pDPj = 1. Then, the dispersion probability is given by:

PDP =JXmax j=0

j Npairs

!

pDPj (24)

This multinomial formulation is very similar to that of the delay threshold probability given in Eq. 6. We again have a multiple bin probability structure, where bin j corresponds to j pairs that are dispersed by more than in steady state. The exhaustive solution to Eq. 24 requires (NVBR)TSP runs and is also infeasible for realistic systems.

5.3 Monte Carlo Simulation

Using Monte Carlo simulation and an approach similar to the ones in 14, 10], we form an unbiased estimate of PDP as follows:

^PDP = 1_N_pairsJXmax

j=1 j^pDPj (25)

where ^pDPj are estimates of the individual probabilities pDPj and are given by:

^

pDPj = 1_N_MC

NXMC

s=1 ;j(s) (26)

where NMC is the total number of Monte Carlo simulation runs and ;j(s) is an indicator

function forj cells dispersing by more than at run s. The estimate of the estimator variance is given by:

^2_{( ^}_P_DP_{) =} 1

(Npairs)2 JXmax

j=1 j

2_^2_(^_p_DP

j) (27)

where ^2_(^_p_DP_j_{) are the estimates of the estimator variances for the individual bins and are}

given by:

^2_(^_p_DP

j) = _N_MC_(N1_MC ;1)

NXMC

s=1(;j(s)

;p^DPj)

2 ₍₂₈₎

Following 14, 10], we generate condence intervals using the result that the condence interval of a weighted sum of individual multinomial probabilities follows a 2 _{distribution.}

5.4 Importance Sampling Method

In ATM networks, the probability of high dispersion should be kept very low, typically below 10;9. As mentioned before, Monte Carlo simulation is not feasible at these probabilities for

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 11 function (pdf) fV1V2(v1v2) to f

V1 V2(v

1v

2) such that the estimate is formed with this new

pdf. Here, V1 refers to the set of all possible connection starting-slot vectors v1 for the

VBR source and V2 refers to the set of all possible connection starting-slot vectors v2 for

the CBR source. Note that V = V1 V2. Let us call the new estimate ^P

DP. We require

that the variance of the new estimate be reduced for a given number of simulation runs, or equivalently, that the number of simulation runs required to achieve a given variance be reduced.

Note that due to the independence of the VBR and CBR connections, the original pdf can be written as: fV1V2(v1v2) = fV1(v1)

fV

2(v2). Since we have a xed CBR source,

this can further be reduced to fV1V2(v1v2) = fV1(v1). Furthermore, since we only bias the

VBR source, f

V1 V2(v

1v

2) = f

V1(v

1). To keep ^PDP unbiased, each important event must

be appropriately weighted or \unbiased". This weight is given by w(v

1). Additionally, we

require that f

V1(v

1)> 0 whenever fV1(v1)> 0.

The important region can be identied by a subset VDP of V such that all vectors in

VDP result in at least one CBR pair dispersed by more than in steady state. For realistic

systems, jVDPjjVj. Monte Carlo simulation is inecient since it samples fromV . We do

not knowVDP a prioriand instead sample from a setV which is known to contain V

DP and

for which jV

jjVj. Note thatV

may contain vectors that cause no dispersion. We form

the unbiased estimate ^P

DP as a linear combination of individual estimates ^p

DPj as before:

^P

DP =JXmax j=1 j Npairs ! ^ p

DPj (29)

For the IS case, the individual estimates ^p

DPj are as follows:

^ p

DPj = 1n_j

nj

X

s=1;

j(s)w

j(s) (30)

where ;

j(s) is the indicator function for pairs dispersed by more than for run s under the

biased pdf f

V1(v

1) andw

j(s) = fV(v(s))=f

V (v(s)) is the IS weight for the vector drawn in

run s. The variance of the IS estimate is found as in Eq. 27, where the individual estimates ^2_(^_p

DPj) are calculated as follows:

^2_(^_p

DPj) = n_j(n_j1;1)

nj

X

s=1(;

j(s)w

j(s);p^

DPj)2 (31)

The improvement factors are calculated as in Section 4 as follows: ^Rnet =

P_J_max

j=1 j2p^

DPj(1;p^

DPj)=nj

P_J_max

j=1 j2^2(^p

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 12 00 00 00 11 11 11 000 000 000 111 111 111 00 00 00 11 11 11 00 00 00 11 11 11 000 000 000 111 111 111 000 000 000 111 111 111 000 000 000 111 111 111 000 000 111 111 000 000 111 111 000 000 111 111 00 00 11 11 00 00 11 11 00 00 00 11 11 11 VBR cells 000 000 000 111 111 111 00 00 00 11 11 11 00 00 00 11 11 11 000 000 000 111 111 111 00 00 00 11 11 11 00 00 00 11 11 11 000 000 000 111 111 111 000 000 000 111 111 111 000 000 111 111 000 000 111 111 00 00 11 11 00 00 11 11 00 00 11 11 000 000 111 111 0 0 0 1 1 1 0 0 0 1 1 1 i i+1 i i+1 VBR cells

Figure 6: Two dierent scenarios in which dispersion can occur.

5.5 IS Biasing Procedure

The function of the biasing procedure is to generate the space V from which the VBR

connection starting-slots are chosen. The simulation algorithm, to be presented in Section 5.6, involves selection of the starting-slots of the VBR sources from V so as to cause a

dispersion of more than for at least one CBR pair. The simulation algorithm uses the results of the biasing procedure to drive the simulation.

Once one CBR pair is forced to disperse by more than , the remaining (if any) VBR sources are selected so as to capture all cases for which there are multiple pairs that disperse by more than .

As shown in Eq. 21, dispersion occurs when the dierence of the departure times of two consecutive CBR cells,Di+1;Di exceeds the specied jitter target . If no VBR cells arrive

between the arrival of the two CBR cellsi and i+1, then there will be no dispersion. Hence, dispersion is caused by the arrival of background VBR cells in between two CBR cells. There are two dierent ways in which two CBR cells i and i+1 can disperse, as illustrated in Fig. 6. In the rst case, the queue empties after the departure of cell i, but before the arrival of cell i + 1, as shown in the top half of Fig. 6. In this case, dispersion is caused by VBR cells that arrive after CBR celli has departed from the queue, but before CBR cell i+1 has departed. In the second case, CBR cells i and i + 1 are both inside the queue for a nonzero amount of time, as shown in the bottom half of Fig. 6. in this case, dispersion is caused by all VBR cells that arrive after the arrival of CBR cell i, but before the arrival of CBR cell i + 1. For Cases 1 and 2, a dierent number of VBR cell arrivals are necessary to cause dispersion. We subsequently analyze these cases separately.

5.5.1 Case 1

For this case, CBR celli departs from the queue before the arrival of cell i+1. An illustration of this case is shown in Fig. 7 in slotted-time, where, without loss of generality, we assume that cell i arrives at slot 0. Also, assume initially that Q0 = 0 before the arrival of cell i.

The CBR cell i exits the buer in e =d1=e slots, as shown in Fig. 7. Then, for case 1, all

VBR cells must arrive within a support l, illustrated in Fig. 7, where 1lTCBR;d1=e.

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cbr

0000000000000 0000000000000

1111111111111 1111111111111

T

i departs

0 0 0

1 1 1

i i+1

VBR cells

support l Q=0

i+1 departs

Figure 7: An illustration of Case 1 in slotted-time.

have no eect on the dispersion. Also note that, by denition, the queue length cannot drop to 0 during a given support l since otherwise, the starting-slot conguration would conform to a dierent support l0< l.

The queue length before the arrival of CBR celli + 1 is given by:

Q = Vcells;l (33)

where Vcells is the total number of VBR cells that arrive within the support l. Hence, CBR

cell i + 1 exits the buer in d(Q + 1)=e slots. To cause a dispersion of more than , we

require that: &

Q + 1

'

;e > (34)

Using the relation that dae > b ) a > b and combining Eq.'s 33 and 34, we nd that the

number of VBR cells in the support of l must satisfy:

Vcells> ( + e);1 +l (35)

Thus, in a support of l where 1 l TCBR ;d1=e, the minimum number of VBR cells

required to cause a dispersion of more than is given by:

Vmin =b( + e);1 +lc+ 1 , for 1l TCBR;d1=e (36)

where e =d1=e slots.

In the above, we assumed that Q0 = 0. If Q0 > 0, then we would have that e0 = d(Q0+ 1)=e > e and from Eq. 36, we would have that V

0

min > Vmin. This means that

stating Vmin as in Eq. 36 encompasses all cases, including the ones for which Q0 > 0.

5.5.2 Case 2

For this case, there is a nonzero amount of time for which CBR cells i and i + 1 are si-multaneously inside the queue. An illustration of this case is shown in Fig. 8 in slotted-time, where we again assume that cell i arrives at slot 0. Again, we assume initially that Q0 = 0 before the arrival of cell i. Hence, we have e = d1=e. Here, the queue length

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cbr

000000000000000000 000000000000000000

111111111111111111 111111111111111111

T

i+1 departs i+1

VBR cells

Q>0 i

support l

i departs

Figure 8: An illustration of Case 2 in slotted-time.

can also contribute to dispersion. For l TCBR, the total amount of service between the

two cells is TCBR regardless of the support and the cell i also contributes by 1= to the

dispersion. Proceeding as in Case 1, we obtain: Vcells>

+ e;

1

!

;1 +TCBR (37)

Note that for Case 2, Vcells is independent of the support. If Q0 > 0, we would have

e0=

d(Q0+ 1)=e and would replace 1= by (Q0+ 1)= in the above expression. Thus, for

Vcells to encompass all dispersive cases, we make the following modication:

V0

cells=

+ e;

1 ;1

!

;1 +TCBR (38)

where e = d1=e. When the support is l > TCBR, Vcells is augmented by l ; TCBR to

compensate for the cells that remain to the left of CBR cell i and hence that have no eect on the dispersion. Thus, for Case 2, Vmin is given by:

Vmin =

8 > > < > > :

j

+ e;

1 ;1

;1 +TCBR

k

+ 1 , TCBR;e + 1 l TCBR

j

+ e;

1 ;1

;1 +TCBR

k

+ 1 +l;TCBR , TCBR < lTCBR+ ^BVBR;1

(39) For Cases 1 and 2, Eq.'s 36 and 39 specify the minimum number of VBR cells necessary for a support of l. The minimum number of VBR connections to provide Vmin cells is then

given by:

Nmin =

&

Vmin

minfl ^BVBRg

'

(40) The maximum number of VBR cells available between CBR cells i and i + 1 is given by:

Vmax =

(

NVBRminfl ^BVBRg , 1 l TCBR

NVBRminfl ^BVBRg;(l;TCBR) ,TCBR < lTCBR+ ^BVBR;1

(41) For a support l, unless Vmax Vmin, there will be no cases which cause a dispersion of more

than . Hence, the minimum number of VBR sources required to cause at least one CBR pair to disperse by more than is given by:

V_minvbr₌ min

1lTCBR+ ^BVBR;1

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Nmin-1

f 0

i

00 00 11 11

00 00 11 11 000 000 111 111 00 00 11 11 00 00 11 11 00 00 11 11 000 000 111 111

support l

i+1

Figure 9: Calculation of the distance f0.

For a support of l, one burst is always xed at slot TCBR ;l. This is also the leftmost

burst. The number of additional VBR cells required in the support l to cause dispersion is given by:

Vextra=Vmin;minfl ^BVBRg (43)

Given a support l and a number of VBR bursts NVBR = Nmin, unless all VBR bursts

arrive within a certain number of slots of the xed burst, there will not be enough cells to cause dispersion. Let f0 be the maximum distance from the xed burst within which all

VBR bursts must arrive so that at least Vmin cells arrive in the support l. The maximum

value of f0 can be found by placing Nmin;1 VBR bursts at slot ^BCBR;l (the beginning of

the burst) and nding the maximum distance for burst Nmin, as seen shaded diagonally in

Fig. 9. The resulting distance is given as follows: f0 =s;

l

Vextra;(Nmin;1)minfl ^BVBRg

m

(44) So, at NVBR = Nmin VBR connections and at the support l, unless all VBR bursts start

within f0 slots of each other, the CBR pairsi and i + 1 will not be dispersed.

At NVBR = Nmin +r connections, where r > 0, each one of the extra VBR bursts can

compensate for one move of the diagonally shaded burst in Fig. 9 to the right. Ifr ^BVBR,

the r extra bursts compensate for the entire diagonally shaded burst and hence no more than Nmin+r bursts are necessary. We have:

fr r>0 =

(

s , if f0 s;r

s;1 , otherwise

(45) Thus, at NVBR =Nmin+r connections and at the support l, unless Nmin VBR bursts start

withinf0 slots of each other orNmin+1 VBR bursts start withinf1 slots of each other, etc.,

the CBR pairs i and i + 1 will not be dispersed.

For each support l for a given CBR pair, the biasing procedure identies the values Nmin and fr, 0 r NVBR;Nmin. It also identies V

vbr

min, the minimum number of VBR

connections necessary to cause a dispersion for any pair.

5.6 Simulation Algorithm

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biasing procedure

simulation algorithm

simulation runs

system parameters

probability IS

estimate draw

vector

vector, weight set V *

Figure 10: Relationship between the simulation procedures.

IfV_minvbr_{> N}_VBR_{, the algorithm exits. Otherwise, it rst calculates a value}_r_over ₌_N_VBR;

V_minvbr_{, which species the number of connections that can be sampled uniformly from the}

entire period at the beginning since only V_minvbr _{bursts are necessary to cause a pair to be}

dispersed. The IS weight is initializedto 1. Subsequently, the algorithm goes through all pairs and all supports to determine which pairs can potentially disperse and which supports within the pairs can potentially cause dispersion. Pairs and supports that can cause dispersion are marked, the others are unmarked. For marked pairs and supports, this results in min and max ranges for which identify the starting slots within which the next burst must be chosen to potentially cause a dispersion. The overall range is given by:

(MINMAX) =

(marked pairsp)

(marked supportsl)(minmax) (46)

This concludes the update phase. Subsequently, the next burst is sampledwithin (MINMAX) and the IS weight is multiplied by (MAX ;MIN)=TSP. The algorithm proceeds with

up-date and sample phases until all bursts for the given vector are sampled. If at a point in the algorithm it is observed that for a given pair and a given support that Nmin bursts arrive

within f0 or Nmin + 1 bursts arrive within f1 etc., a Monte Carlo ag is set, meaning that

a potential dispersion has already occurred and that the remaining bursts should be sam-pled uniformly from TSP. This facet enables the algorithm to capture all vectors that cause

multiple pairs to disperse as well.

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simulation procedure:

roverNVBR;Vminvbr.

if (r_over<0) exit. /* not enough VBR connections */

w=1. /* initialize IS weight */

v =(zeros(NVBR)j0). /* initialize the vector */ unmark MC. /* initialize Monte Carlo flag */

for (burst=1 burst <= r_over burst++) /* the first r bursts are MC */

{ sample v(burst) uniformly over 0TSP;1].

call update procedure (vMINMAX) /* this generates MIN and MAX */

for (burst=r+1 burst <= NVBR burst++) /* sample other bursts */

{ sample v(burst) uniformly over MINMAX].

{ ww(MAX;MIN)=TSP.

{ call update procedure (vMINMAX) /* update after every burst */

return v, w. /* return the vector and IS weight */ update procedure(vMINMAX):

for (pair=1 pair <= T_SP=T_CBR pair++) /* all pairs */

{ for (support=1 support <= TCBR+ ^

BVBR;1 support++) /* supports */ if (pair can be dispersed at support) mark (pair@support).

else unmark (pair@support)

update min and max from v.

for (r=0 r <= NVBR;Nmin r++) /* check for MC */ if (Nmin+r bursts are within fr) mark MC.

else unmark MC.

if (MC) (minmax)(0TSP;1)

(MINMAX)

S

(minmax) /* update MIN and MAX */

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5.7 Specication of Dispersion

As mentioned in Section 2, we specify the dispersion target in terms of a percentage of the CBR period TCBR. There is a maximum dispersion that can be specied for a given system

and a given number of background VBR sources.

Due to the physical constraints of the system, for there to be dispersion in steady state, all cells must exit within a superperiod TSP. Otherwise, all of the CBR cells will be clumped

at departure. The physical constraint implies the following:

&

1

'

+ + (Npairs;2)

&

1

'

TSP;TCBR (47)

which results in the following physical upper bound for dispersion: phys=TSP;(Npairs;1)

&

1

'

;TCBR (48)

We also know from Section 5.5 that givenNVBR VBR connections, the dispersion for a given

support l satises:

max(l)

1

NVBRminfl ^BVBRg+ 1;l

;

&

1

'

(49) Hence, an upper bound for the dispersion in terms of the system parameters is given by:

sys= max_l l(l) (50)

Hence, for the simulation runs, the target dispersion value must satisfy:

max= minfphyssysg (51)

No dispersion value above max is possible.

6 Experimental Results

6.1 Two-Point CDV Results

For the two-point cell delay variation, we consider the same systems considered in 10]. The input parameters for the sources are listed in Table 1. System A consists of CBR sources and VBR-1 sources. System B consists of CBR sources and VBR-2 sources. The system parameters and derived parameters for these systems are listed in Table 2.

Here, we made the assumption that the two classes of connections occupied consecutive lines and that the CBR connections occupied the rst lines. Hence, the CBR connections arrive at lines 1 through NCBR and the VBR connections arrive at lines NCBR + 1 through

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Source ^, , ^B, T,

Mbps Mbps cells slots

CBR-1 10 10 1 1

VBR-1 50 5 25 250

VBR-2 500 2 50 12,500

Table 1: Sources for the 2-Point CDV.

System Sources ,^ K, TSP, ,

Mbps cells slots cells/slot

A CBR-1, VBR-1 120 200 250 2.4

B CBR-1, VBR-2 120 200 12,500 0.24

Table 2: System parameters and derived parameters for the 2-Point CDV.

HOL priority which would be implemented at the servers). The simulations were run by xing the number of VBR sources to the minimum required for dispersion when mixed with a single CBR source. We then varied the number of CBR sources. Had we varied the number of VBR sources, we would have obtained curved as in 14] due to the dominant nature of the VBR class.

We rst considered System A with two threshold levels = 40%and = 50%, corre-sponding to queue lengths of 90 and 110 cells, respectively. The number of VBR sources was set at 6 and the number of CBR sources was varied. The simulation stopping condition was set at 100 hits per bin. The dispersion estimates are plotted in Fig. 11. The aggregate 2-Point CDV estimates (increasing curves) and the improvement factors (decreasing curves) are plotted as solid lines. The individual estimates for the CBR and VBR trac are plotted as dotted and dashed lines, respectively. The threshold levels = 40%and = 50%are rep-resented by circle and square points, respectively, and the condence intervals are depicted with the same type point as the corresponding threshold.

For = 50%, at least 2 CBR sources were necessary for there to be dispersion. The condence intervals and improvement results for the aggregate dispersion were discussed in 10]. We observe from Fig. 11 that the improvement is inversely proportional to the probability being estimated.

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1 2 3 4 5 6 7

1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x10−10 1x10−9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3

number of CBR connections -3

-4 -5 -6 -7 -8 -9

-11 -12 -13 -14 -15 -10

0 2

1

3

log(2-point CDV estimate)

4 6 7 8

1 9 10

5

2 3 4 5 6 7

11

log(improvement factor)

tau=40%

tau=50%

Figure 11: 2-Point CDV estimates for system A, 6 VBR sources, number of CBR sources varying (increasing solid lines are aggregate 2-Point CDV's, decreasing solid lines are im-provement factors, dashed lines are for VBR, dotted lines are for CBR. is for = 40%, 4 is for = 50% ).

the average queue length will increase and the eect of the line priority will diminish. Next, we considered System B with two threshold levels = 85%and = 95%, corre-sponding to queue lengths of 170 and 190 cells, respectively. The number of VBR sources was set at 4 and the number of CBR sources was varied. The dispersion estimates are plotted in Fig. 12. The threshold levels = 85% and = 95%are represented by circle and square points, respectively, and the condence intervals are depicted with the same type point as the corresponding threshold. For = 95%, at least 2 CBR connections were required to cause dispersion.

The results obtained for System B are similar to that of System A. Again, the improve-ment factors are inversely proportional to the probability being estimated. Also, the 2-Point CDV for the CBR trac is always lower than that of the aggregate trac and vice-versa for the VBR trac. However, due to the large period of System B, the addition of CBR sources does not have as large an eect on the dispersion. For the same reason, convergence of the three curves is not as apparent for System B as it was for System A.

7 One-Point CDV Results

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1 2 3 4 5 6 7

1x10−14 1x10−13 1x10−12 1x10−11 1x10−10 1x10−9 1x10−8 1x10−7 1x10−6 1x10−5

1 3 5 6 7

-13 -12 -11 -10 -9 -8 -7 -6

2 -5

4

2 3 4 5 6 7 8 9 10 11

number of CBR connections

tau=85%

tau=95%

-14

Figure 12: 2-Point CDV estimates for system B, 4 VBR sources, number of CBR sources varying (increasing solid lines are aggregate 2-Point CDV's, decreasing solid lines are im-provement factors, dashed lines are for VBR, dotted lines are for CBR. is for = 85%, 4 is for = 95% ).

and F are generated by varying a single VBR parameter of System C and Systems H, I and J are generated by varying a single VBR parameter of System G. System C has < 1 and System G has > 1. The CBR source is identical for all cases.

Here, we assumed that there is a single CBR source at line 1 and the VBR sources occupy the remaining lines. In this manner, the CBR source is the observed source and the VBR sources constitute the background trac. The simulations were run by xing the number of VBR sources and varying the target 1-Point dispersion parameter from 100%of TCBR

to max. For systems C-F, NVBR = 10 and for Systems G-J, NVBR = 14. The simulation

stopping condition was set at 200 hits per bin.

The simulation results for Systems C and D, which vary in the mean rate of the VBR source, are plotted in Fig. 13. The mean VBR rates are given by 10 Mbps and 5 Mbps for Systems C and D, respectively. In Fig. 13, the decreasing curves are the 1-Point CDV estimates and the increasing curves are the improvement factors. the curves for System C are solid and marked with circular points, whereas those for System D are dashed and marked with triangular points. For each case, the condence intervals are marked with the same style point as the corresponding 1-Point CDV curve. The 1-Point CDV curves are initially almost linear (in log scale) and then exhibit an asymptotic behavior until the maximum dispersion value max. A dispersion value abovemax is not possible. The points enclosed in

circles at the rightmost value of the CDV curves represent the fact that for that given and for a given pair, there is only one valid support and for that support f0 = 0. Thus, the

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Source ^, , ^B, T,

Mbps Mbps cells slots

CBR-2 5 5 1 1

VBR-3 500 10 20 1000

VBR-4 500 5 20 2000

VBR-5 250 10 20 500

VBR-6 500 10 10 500

VBR-7 100 5 10 200

VBR-8 100 2 10 500

VBR-9 200 5 10 400

VBR-10 100 5 8 160

Table 3: Sources for the 1-Point CDV.

System Sources ,^ K, TSP, ,

Mbps cells slots cells/slot

C CBR-2, VBR-3 120 200 1000 0.24

D CBR-2, VBR-4 120 200 2000 0.24

E CBR-2, VBR-5 120 200 500 0.48

F CBR-2, VBR-6 120 200 500 0.24

G CBR-2, VBR-7 120 200 200 1.2

H CBR-2, VBR-8 120 200 500 1.2

I CBR-2, VBR-9 120 200 400 0.6

J CBR-2, VBR-10 120 200 160 1.2

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 1x10−34

1x10−33 1x10−32 1x10−31 1x10−30 1x10−29 1x10−28 1x10−27 1x10−26 1x10−25 1x10−24 1x10−23 1x10−22 1x10−21 1x10−20 1x10−19 1x10−18 1x10−17 1x10−16 1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x101x10−10 −9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3 1x10−2 1x10−1

1 2 3 4 5 6 7 8 9

-34 -31 -28 -25 -22 -19

sys D

-16 -13 -10 -7 -4

-1 21

18

15

12

9

6

3

0

dispersion x 100%

sys C

Figure 13: 1-Point CDV estimates for Systems C and D, 10 VBR sources w/ varying (decreasing lines are 1-Point CDV's, increasing lines are improvement factors. and solid

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 1x10−31

1x10−30 1x10−29 1x10−28 1x10−27 1x10−26 1x10−25 1x10−24 1x10−23 1x10−22 1x10−21 1x10−20 1x10−19 1x10−18 1x10−17 1x10−16 1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x101x10−10 −9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3 1x10−2 1x10−1

1 2 3 4 5 6 7 8 9

-31 -28 -25 -22 -19

sys E

-16 -13 -10 -7 -4

-1 ₁₉

16

13

10

7

4

1

dispersion x 100% sys C

Figure 14: 1-Point CDV estimates for Systems C and E, 10 VBR sources w/ ^ varying (decreasing lines are 1-Point CDV's, increasing lines are improvement factors. and solid

lines are for System C, 4 and dashed lines are for System E).

1-Point CDV probability can be found exactly using:

PDP = _(T_SP₎1_NVBR (52)

For the exact points, the improvement generated is many orders of magnitude larger than for the other points and hence is not shown in Fig. 13.

As seen from Fig. 13, the improvement factors are again inversely proportional to the 1-Point CDV probability being estimated. The 1-Point CDV is higher for System C, since System C has a higher mean VBR rate. The dierence between the curves increases as is increased form 100%until it becomes constant as the asymptote is approached.

Systems C and E vary in the VBR peak rate and Systems C and F vary in the VBR burst length. The simulation results for these systems are plotted in Fig.'s 14 and 15, respectively. For both cases, we once again observe that the improvement in simulation eciency is inversely proportional to the probability being estimated. We also observe similar properties for the 1-Point CDV curves.

For System C, ^VBR = 500 Mbps, compared to ^VBR = 250 Mbps for System E. From

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 1x10−31

1x10−30 1x10−29 1x10−28 1x10−27 1x10−26 1x10−25 1x10−24 1x10−23 1x10−22 1x10−21 1x10−20 1x10−19 1x10−18 1x10−17 1x10−16 1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x101x10−10 −9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3 1x10−2 1x10−1

1 2 3 4 5 6 7 8 9

-31 -28 -25 -22 -19 -16

sys F

-13 -10 -7 -4

-1 19

16

13

10

7

4

1

sys C

Figure 15: 1-Point CDV estimates for Systems C and F, 10 VBR sources w/ ^B varying (decreasing lines are 1-Point CDV's, increasing lines are improvement factors. and solid

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

1x10−39 1x10−38 1x10−37 1x10−36 1x10−35 1x10−34 1x10−33 1x10−32 1x10−31 1x10−30 1x10−29 1x10−28 1x10−27 1x10−26 1x10−25 1x10−24 1x10−23 1x10−22 1x10−21 1x10−20 1x10−19 1x10−18 1x10−17 1x10−16 1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x101x10−10 −9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3 1x10−2

1 2 3 4 5 6

-39 -35 -31 -27 -23 -19 -15

sys H -11

-7

-3 33

29

25

21

9 17

13

5

1

sys G

Figure 16: 1-Point CDV estimates for Systems G and H, 14 VBR sources w/ varying (decreasing lines are 1-Point CDV's, increasing lines are improvement factors. and solid

lines are for System G, 4 and dashed lines are for System H).

cells, System F had ^BVBR = 10 cells. The 1-Point CDV for System F was signicantly lower

than that for System C. Also, themax for System F was less than half of that for System C.

The 1-Point CDV probability estimates and improvement factors for Systems G and H, G and I, G and J are plotted in Fig.'s 16-18. For Systems G-J, the service rate is greater than that for Systems C-F. The 1-Point CDV curves exhibit the same asymptotic behavior and the improvements are again inversely proportional to the probability being estimated.

However, when is increased, more VBR sources are necessary to cause dispersion. Hence, we have 14 VBR sources for Systems G-J. Also, maxand the 1-Point CDV

probabil-ities decrease as is increased. Since the probabilities decrease, we also observe an increase in the improvement factors.

The general behavior of the curves when the VBR parameters are varied is similar to those of Systems C-F. However, for Systems G-J, greater changes in the given parameters are required to cause changes similar to the ones in Systems C-F for the 1-Point CDV probabilities.

8 Conclusion

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1x10−38

1x10−37 1x10−36 1x10−35 1x10−34 1x10−33 1x10−32 1x10−31 1x10−30 1x10−29 1x10−28 1x10−27 1x10−26 1x10−25 1x10−24 1x10−23 1x10−22 1x10−21 1x10−20 1x10−19 1x10−18 1x10−17 1x10−16 1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x101x10−10 −9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3 1x10−2

4 5 6

3 2

1 -38 -34 -30 -22

-26 -18 -14

sys I

-10 -6

-2 29

25

21

17

13

9

5

1

dispersion x 100% sys G

Figure 17: 1-Point CDV estimates for Systems G and I, 14 VBR sources w/ ^ varying (decreasing lines are 1-Point CDV's, increasing lines are improvement factors. and solid

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

1x10−33 1x10−32 1x10−31 1x10−30 1x10−29 1x10−28 1x10−27 1x10−26 1x10−25 1x10−24 1x10−23 1x10−22 1x10−21 1x10−20 1x10−19 1x10−18 1x10−17 1x10−16 1x10−15 1x10−14 1x10−13 1x10−12 1x10−11 1x101x10−10 −9 1x10−8 1x10−7 1x10−6 1x10−5 1x10−4 1x10−3 1x10−2

1 2 3 4 5 6

-33 -30 -24 -21 -18 -15 -9 -6

-3 28

25

22

19

16

13

10

7

4

1

sys G

sys J

-27 -12

Figure 18: 1-Point CDV estimates for Systems G and J, 14 VBR sources w/ ^B varying (decreasing lines are 1-Point CDV's, increasing lines are improvement factors. and solid

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E cient Simulation of Rare Jitter Probabilities in ATM Switches..., A. A. Akyamac 29 networks. We used the ATM Forum standardized connection trac descriptors to charac-terize the input trac (operational approach) and we considered dispersion as a measure of CDV.

We developed ecient simulation methods to estimate the rare tail dispersion proba-bilities for both cases. In each case, we developed multinomial formulations to eectively remove correlations (due to bursty trac) between important events. We extended our pre-vious work utilizing Importance Sampling in the case of delay threshold probabilities for heterogeneous trac to estimate the 2-Point CDV probabilities. Subsequently we developed a new ecient simulation method using Importance Sampling to estimate the 1-Point CDV probabilities.

For the experimental systems considered for both cases, we observed that the improve-ment in simulation eciency (speedup over standard Monte Carlo simulation) was inversely proportional to the probability being estimated.

References

1] ATM User-Network Interface Specication, Version 3.0. The ATM Forum, September 10, 1993. The ATM Forum.

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